# Factorization systems in a triangulated/stable category

Google is of little help when questioned about "factorization systems in a triangulated category". Are there informations about them? Are there "natural" (meaning: "obviously defined") OFS/WFS in a triangulated category or (better) in a stable $\infty$-category?

(Presenting the latter as a model category $\cal M$, OFS in $\operatorname{ho}\cal M$ are in correspondence with homotopy factorization systems in $\cal M$, and presenting it via a stable quasicategory $\cal C$ they correspond to quasicategorical factorization systems on $\cal C$. Is this point of view of any help?)

I tried to elaborate on the good old $(\textrm{Mono}, \textrm{Epi})$ or $(\textrm{Epi}, \textrm{Mono})$ factorizations, but any sensible definition of "epimorphism" and "monomorphism" in a triangulated/stable category turns out to be trivial in some sense (I can elaborate on this if necessary).

• What should it be in e. g. the stable homotopy category? I cannot think of anything sensible: in a given stable homotopy class there are too many representatives of various kinds - fibrations, cofibrations, ... – მამუკა ჯიბლაძე May 1 '14 at 9:48
• I'm trying to go in the same direction, indeed. I'm led to think that in the context of stable homotopy the notion of orthogonal factorization system is unsuited; maybe one have to replace it with the notion of "homotopy factorization system" : math.ku.dk/~alexb/transport.pdf section 2 (they took the def. from Joyal, I think). The stable homotopy category should be one of the simplest examples, as it is in some sense the "free stable category on the point". – Fosco May 1 '14 at 10:27
• I don't know the answer, but have you looked into Emily Riehl's book on Categorical Homotopy Theory math.harvard.edu/~eriehl/266x? Her thesis dealt very carefully with weak factorization systems in model categories and she has been active in $\infty$-categories lately. So it seems likely to me that she'd have thought about this question at some point. – David White May 5 '14 at 1:40
• This nforum discussion should probably be linked to here. – user62675 Feb 2 '15 at 1:23